cp's OEIS Frontend

A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

%I A380217 #10 Jan 21 2025 13:11:12
%S A380217 1,15,42,54,75,77,95,99,100,132,182,195,221,234,245,253,290,312,315,
%T A380217 329,350,357,405,420,423,437,450,459,476,494,510,540,555,559,560,612,
%U A380217 627,665,715,720,740,798,816,833,854,855,858,893,897,899,979,1026,1064
%N A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.
%C A380217 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum and product of prime indices are A056239 and A003963 respectively.
%F A380217 a(n) = A379319(n)/2.
%e A380217 The prime indices of 75 are {2,3,3}, with product 18 and sum 8, and since 18 is a multiple of 8+1, 75 is in the sequence.
%e A380217 The terms together with their prime indices begin:
%e A380217      1: {}
%e A380217     15: {2,3}
%e A380217     42: {1,2,4}
%e A380217     54: {1,2,2,2}
%e A380217     75: {2,3,3}
%e A380217     77: {4,5}
%e A380217     95: {3,8}
%e A380217     99: {2,2,5}
%e A380217    100: {1,1,3,3}
%e A380217    132: {1,1,2,5}
%e A380217    182: {1,4,6}
%e A380217    195: {2,3,6}
%e A380217    221: {6,7}
%e A380217    234: {1,2,2,6}
%e A380217    245: {3,4,4}
%t A380217 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380217 Select[Range[100],Divisible[Times@@prix[#],1+Total[prix[#]]]&]
%o A380217 (PARI) vpind(n)=my(v=List(), f=factor(n)); for(i=1, #f~, for(j=1, f[i, 2], listput(v, primepi(f[i, 1])))); Vec(v); \\ A112798
%o A380217 isok(k) = my(vind = vpind(k)); (vecprod(vind) % (vecsum(vind)+1)) == 0; \\ _Michel Marcus_, Jan 21 2025
%Y A380217 The case of equality is A325041, counted by A380218 = A028422 except n=3.
%Y A380217 Without "plus one" we get A326149, counted by A057568, see A379733, A379734, A379735.
%Y A380217 Double all terms to get A379319.
%Y A380217 Partitions of this type are counted by A379320.
%Y A380217 Counting and ranking multisets by comparing sum and product:
%Y A380217 - same: A001055, ranks A301987
%Y A380217 - multiple: A057567, ranks A326155
%Y A380217 - divisor: A057568, ranks A326149
%Y A380217 - greater than: A096276 shifted right, ranks A325038
%Y A380217 - greater or equal: A096276, ranks A325044
%Y A380217 - less than: A114324, ranks A325037, see A318029, A379720
%Y A380217 - less or equal: A319005, ranks A379721, see A025147
%Y A380217 - different: A379736, ranks A379722, see A111133
%Y A380217 Cf. A069016, A301988, A318950, A319000, A324851, A379318, A379671, A379844, A379845.
%K A380217 nonn
%O A380217 1,2
%A A380217 _Gus Wiseman_, Jan 18 2025